Let r be the binomial random variable corresponding to the number of people that will live beyond their 90th birthday, r ≥ 15.

We want to find
P(r ≥ 15)
using the normal approximation given 625 trials and a probability of a 4.4% success on a single trial.

P(r ≥ 15) = 0.9943.

Step-by-step explanation:

We use the normal approximation to the binomial to solve this question.

Binomial probability distribution

Probability of exactly x successes on n repeated trials, with p probability.

Can be approximated to a normal distribution, using the expected value and the standard deviation.

The expected value of the binomial distribution is:

The standard deviation of the binomial distribution is:

Normal probability distribution

Problems of normally distributed distributions can be solved using the z-score formula.

In a set with mean and standard deviation , the z-score of a measure X is given by:

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

When we are approximating a binomial distribution to a normal one, we have that , .

625 trials and a probability of a 4.4% success on a single trial.

This means that

Mean and standard deviation:

P(r ≥ 15)

Using continuity correction, this is , which is 1 subtracted by the p-value of Z when X = 14.5. So

has a p-value of 0.0057

1 - 0.0057 = 0.9943

So

P(r ≥ 15) = 0.9943.

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step-by-step explanation: